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The group of parallel axes about which the rigid body Dscillates with the sa-me period forms two coaxial circular cylinders. Fig. 140, whose common axis passes through the cen-
FIG. 139.
fcer of mass.   The cylinders which correspond to the minimum value of the period coincide and have a common radius K.
1.   Find the period of the following physical pendulums:
(a)  A uniform rod, the transverse dimensions of which are negligible compared with the length, oscillates about a horizontal axis through one 3nd.
(b)  A sphere suspended from a horizontal axis by means of a string of negligible mass.    Discuss the changes in the period as the axis approaches, the center of the sphere.
(c)  A circular flat ring oscillates about an axis which forms an element of the inner surface.
(d)  A door oscillates about the line of the hinges which make an angle a with the vertical.
2.   A sphere of radius a oscillates back and forth in a perfectly smooth spherical bowl of radius &.   Find the period of oscillation.   The sphere is supposed to have no rolling motion.
3.   What effect on the period of a pendulum would be produced by a change in the mass of the bob, or of the length of the string, or in the radius of the earth, or in the length of the day, or in the latitude of the, location?
4.   A seconds pendulum loses 30 seconds per day at the summit of & mountain.   Find the height of the mountain, considering the earth to be